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\title{The 59th William Lowell Putnam Mathematical Competition \\
    Saturday, December 5, 1998}
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\begin{itemize}
\item[A--1] 
A right circular cone has base of radius 1 and height 3.  A
cube is inscribed in the cone so that one face of the cube is
contained in the base of the cone.  What is the side-length of
the cube?

\item[A--2]
Let $s$ be any arc of the unit circle lying entirely in the first
quadrant.  Let $A$ be the area of the region lying below $s$ and 
above the $x$-axis and let $B$ be the area of the region lying to the
right of the $y$-axis and to the left of $s$.  Prove that $A+B$ depends 
only on the arc length, and not on the position, of $s$. 

\item[A--3]
Let $f$ be a real function on the real line with continuous third 
derivative.  Prove that there exists a point $a$ such that 
\[f(a)\cdot f'(a) \cdot f''(a) \cdot f'''(a)\geq 0 .\]
  
\item[A--4]
Let $A_1=0$ and $A_2=1$.  For $n>2$, the number $A_n$ is defined by
concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from
left to right.  For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$,
$A_5=A_4 A_3 = 10110$, and so forth.  Determine all $n$ such that
$11$ divides $A_n$.

\item[A--5]
Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$
whose union contains a set $E\subseteq \mathbb R^2$.  Show that there
is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$
such that 
\[E\subseteq \cup_{j=1}^n 3D_j.\]
Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the
disc of radius $3r$ and center $P$.  

\item[A--6]
Let $A, B, C$ denote distinct points with integer coordinates in $\mathbb
R^2$.  Prove that if
\[(|AB|+|BC|)^2<8\cdot [ABC]+1\]
then $A, B, C$ are three vertices of a square.  Here $|XY|$ is the length
of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

\item[B--1]
Find the minimum value of 
\[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\]
for $x>0$.

\item[B--2]
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a 
triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the
line $y=x$.  You may assume that a triangle of minimum perimeter exists.

\item[B--3]
let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$
the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon
inscribed in $C$.  Determine the surface area of that portion of $H$ lying
over the planar region inside $P$, and write your answer in the form
$A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

\item[B--4]
Find necessary and sufficient conditions on positive integers $m$ and $n$
so that 
\[\sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.\]

\item[B--5]
Let $N$ be the positive integer with 1998 decimal digits, all of them 1;
that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of $\sqrt N$.

\item[B--6]
Prove that, for any integers $a, b, c$, there exists a positive integer
$n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.

\end{itemize}

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